Melt the butter in a skillet over medium heat. Stir in the onion, and cook until softened and translucent, about 5 minutes. Add the celery and bell pepper; continue cooking until the vegetables are tender, about 5 minutes. Stir in the mustard and cayenne pepper; cook 2 minutes. Scrape the mixture into a mixing bowl and refrigerate until cold.
Stir the mayonnaise into the chilled vegetable mixture along with the soft bread crumbs, 1 tablespoon lemon zest, parsley, and beaten egg. Stir until combined, then gently fold in the crab meat. Season to taste with salt and pepper. Pour the dry bread crumbs into a shallow bowl. Form the crab mixture into 8 cakes, 1/2 inch thick. Press the crab cakes into the bread crumbs one at a time until coated on all sides. Gently shake off excess crumbs, and place the crab cakes onto a plate. Cover with plastic wrap and refrigerate 1 hour to overnight.
Prepare the vinaigrette by whisking 3/4 cup vegetable oil with the zest of 1 lemon, lemon juice, ginger, jalapeno, sugar, and salt. Store in the refrigerator until ready to use.
Heat the frying oil in a large skillet over medium-high heat. While the oil is heating, toss the mixed salad greens with the vinaigrette in a mixing bowl, and divide onto 8 serving plates. Arrange the sliced avocado over the salads, and set aside.
Once the oil is shimmering with heat, fry the crab cakes until golden brown on each side, 2 to 3 minutes per side. Briefly drain on a paper towel-lined plate to rid the cakes of excess grease. Place a crab cake on top of each salad and serve immediately.